# Properties

 Label 896.1.h.a Level $896$ Weight $1$ Character orbit 896.h Self dual yes Analytic conductor $0.447$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -56 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$896 = 2^{7} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 896.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.447162251319$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.1568.1 Artin image: $D_8$ Artin field: Galois closure of 8.0.1258815488.2

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} -\beta q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q -\beta q^{3} -\beta q^{5} - q^{7} + q^{9} + \beta q^{13} + 2 q^{15} + \beta q^{19} + \beta q^{21} + q^{25} + \beta q^{35} -2 q^{39} -\beta q^{45} + q^{49} -2 q^{57} + \beta q^{59} -\beta q^{61} - q^{63} -2 q^{65} + 2 q^{71} -\beta q^{75} + 2 q^{79} - q^{81} -\beta q^{83} -\beta q^{91} -2 q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{7} + 2 q^{9} + 4 q^{15} + 2 q^{25} - 4 q^{39} + 2 q^{49} - 4 q^{57} - 2 q^{63} - 4 q^{65} + 4 q^{71} + 4 q^{79} - 2 q^{81} - 4 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$645$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 1.41421 −1.41421
0 −1.41421 0 −1.41421 0 −1.00000 0 1.00000 0
321.2 0 1.41421 0 1.41421 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.1.h.a 2
4.b odd 2 1 896.1.h.b yes 2
7.b odd 2 1 inner 896.1.h.a 2
8.b even 2 1 inner 896.1.h.a 2
8.d odd 2 1 896.1.h.b yes 2
16.e even 4 2 1792.1.c.d 2
16.f odd 4 2 1792.1.c.c 2
28.d even 2 1 896.1.h.b yes 2
56.e even 2 1 896.1.h.b yes 2
56.h odd 2 1 CM 896.1.h.a 2
112.j even 4 2 1792.1.c.c 2
112.l odd 4 2 1792.1.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.1.h.a 2 1.a even 1 1 trivial
896.1.h.a 2 7.b odd 2 1 inner
896.1.h.a 2 8.b even 2 1 inner
896.1.h.a 2 56.h odd 2 1 CM
896.1.h.b yes 2 4.b odd 2 1
896.1.h.b yes 2 8.d odd 2 1
896.1.h.b yes 2 28.d even 2 1
896.1.h.b yes 2 56.e even 2 1
1792.1.c.c 2 16.f odd 4 2
1792.1.c.c 2 112.j even 4 2
1792.1.c.d 2 16.e even 4 2
1792.1.c.d 2 112.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{71} - 2$$ acting on $$S_{1}^{\mathrm{new}}(896, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-2 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-2 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$-2 + T^{2}$$
$61$ $$-2 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$( -2 + T )^{2}$$
$83$ $$-2 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$